ARTICLE

Estimating reach-specific fish movement probabilities in riverswith a Bayesian state-space model: application to sea lampreypassage and capture at damsChristopher M. Holbrook, Nicholas S. Johnson, Juan P. Steibel, Michael B. Twohey, Thomas R. Binder,Charles C. Krueger, and Michael L. Jones

Abstract: Improved methods are needed to evaluate barriers and traps for control and assessment of invasive sea lamprey(Petromyzon marinus) in the Great Lakes. A Bayesian state-space model provided reach-specific probabilities of movement, includ-ing trap capture and dam passage, for 148 acoustic tagged invasive sea lamprey in the lower Cheboygan River, Michigan, atributary to Lake Huron. Reach-specific movement probabilities were combined to obtain estimates of spatial distribution andabundance needed to evaluate a barrier and trap complex for sea lamprey control and assessment. Of an estimated 21 828 –29 300 adult sea lampreys in the river, 0%–2%, or 0–514 untagged lampreys, could have passed upstream of the dam, and 46%–61%were caught in the trap. Although no tagged lampreys passed above the dam (0/148), our sample size was not sufficient toconsider the lock and dam a complete barrier to sea lamprey. Results also showed that existing traps are in good locationsbecause 83%–96% of the population was vulnerable to existing traps. However, only 52%–69% of lampreys vulnerable to trapswere caught, suggesting that traps can be improved. The approach used in this study was a novel use of Bayesian state-spacemodels that may have broader applications, including evaluation of barriers for other invasive species (e.g., Asian carp(Hypophthalmichthys spp.)) and fish passage structures for other diadromous fishes.

Résumé : Il est nécessaire d'améliorer les méthodes d'évaluation des barrières et pièges utilisés pour contrôler et évaluer lalamproie marine (Petromyzon marinus), une espèce envahissante, dans les Grands Lacs. Un modèle bayésien d'espace d'états aproduit des probabilités de déplacement, dont le piégeage et le passage de barrage, dans différents biefs pour 148 lamproiesmarines envahissantes dotées de marques acoustiques dans le cours inférieur de la rivière Cheboygan (Michigan), un affluent dulac Huron. Les probabilités de déplacement propres au bief ont été combinées pour obtenir les estimations de la répartitionspatiale et de l'abondance requises pour évaluer un complexe de barrière et piège pour le contrôle et l'évaluation des lamproies.Sur un nombre estimé de 21 828 a 29 300 lamproies adultes dans la rivière, de 0 % a 2 % des lamproies non marquées, soit de 0a 514 poissons, pourraient être passées en amont du barrage, et de 46 % a 61 % ont été prises dans le piège. Bien qu'aucunelamproie marquée ne soit passée en amont du barrage (0/148), la taille de l'échantillon est trop petite pour permettre d'établir quel'écluse-barrage constitue une barrière complètement étanche au passage des lamproies. Les résultats ont également démontréque les pièges existants sont bien situés, puisque de 83 % a 96 % de la population était vulnérable aux pièges existants. Cependant,de 52 % a 69 % seulement des lamproies vulnérables aux pièges ont été capturées, indiquant que ces pièges peuvent êtreaméliorés. L'approche employée constitue une utilisation novatrice des modèles bayésiens d'espace d'états qui pourrait avoir desapplications plus larges, notamment pour l'évaluation des barrières au passage d'autres espèces envahissantes (p. ex. carpeasiatique (Hypophthalmichthys spp.)) et des passes a poissons pour d'autres espèces diadromes. [Traduit par la Rédaction]

IntroductionThe Laurentian Great Lakes were severely disrupted by the ar-

rival of invasive sea lamprey (Petromyzon marinus) through theWelland Canal in the early 1900s (Smith and Tibbles 1980; Hansen1999). After hatching, nonparasitic, filter-feeding larvae remain instreams for 1–7 years and then transform into parasitic juveniles(Applegate 1950). After transformation, juveniles migrate down-stream and enter lakes where they grow rapidly by feeding on the

body fluid of other fishes. After 12–18 months in the lake, semel-parous adults cease feeding and ascend tributaries to spawn dur-ing spring and early summer. Like other anadromous fishes, sealampreys have been most vulnerable to life cycle interruptions dur-ing their larval stream-resident and stream migratory adult stages,and control strategies have been developed to target those stages.

In the Great Lakes, sea lamprey populations have been sup-pressed with selective pesticides (lampricide) that target larvae instreams, barriers that block access to spawning habitat, and traps

Received 5 November 2013. Accepted 23 June 2014.

Paper handled by Associate Editor Michael Bradford.

C.M. Holbrook and N.S. Johnson. US Geological Survey, Great Lakes Science Center, Hammond Bay Biological Station, 11188 Ray Road, Millersburg,MI 49759, USA.J.P. Steibel. Michigan State University, Department of Animal Sciences, 1205-I Anthony Hall, East Lansing, MI 48824, USA; Michigan State University,Department of Fisheries and Wildlife, Room 7 Natural Resources, East Lansing, MI 48824, USA.M.B. Twohey. US Fish and Wildlife Service, Marquette Biological Station, 3090 Wright Street, Marquette, MI 49855, USA.T.R. Binder. Great Lakes Fishery Commission, 2100 Commonwealth Blvd., Suite 100, Ann Arbor, MI 48105, USA; Michigan State University, Departmentof Fisheries and Wildlife, Room 7 Natural Resources, East Lansing, MI 48824, USA.C.C. Krueger. Great Lakes Fishery Commission, 2100 Commonwealth Blvd., Suite 100, Ann Arbor, MI 48105, USA.M.L. Jones. Michigan State University, Department of Fisheries and Wildlife, Room 7 Natural Resources, East Lansing, MI 48824, USA.Corresponding author: Christopher M. Holbrook (e-mail: cholbrook@usgs.gov).

1713

Can. J. Fish. Aquat. Sci. 71: 1713–1729 (2014) dx.doi.org/10.1139/cjfas-2013-0581 Published at www.nrcresearchpress.com/cjfas on 11 July 2014.

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.

that remove adults prior to spawning (Christie and Goddard2003). Improving effectiveness of traps and barriers, which targetsea lampreys during the adult stage, has been a priority of the sealamprey research program (Jones 2007; McLaughlin et al. 2007).However, empirical evaluations of traps and barriers have beenlimited to indirect evidence of upstream passage (e.g., presence oflarval sea lampreys upstream of a barrier), direct observation inshallow water (Applegate 1950), or mark–recapture methods forwhich critical assumptions have not been rigorously tested.

Barrier construction was among the earliest efforts used to con-trol sea lampreys in the Great Lakes, and a need exists to bothevaluate current barriers and establish new ones (Swink 1999;Lavis et al. 2003; Johnson et al. 2014). The fraction of a populationblocked by a barrier provides an intuitive basis to compare barri-ers, but does not represent the effectiveness of any single barrierbecause upstream recruitment dynamics (e.g., stock–recruitment)often depend on the number of mature individuals above thebarrier (i.e., stock size), not the proportion passing. Thus, thepassage proportion is not sufficient to determine effectiveness ofa barrier as a sea lamprey control device. Instead, the number ofindividuals that escape above a dam should be estimated.

Although the primary function of a barrier is to block access ofadults to upstream spawning habitat, barriers also facilitate re-moval of individuals prior to spawning via barrier-integratedtraps. Little is known about the fate of sea lampreys that encoun-ter an impassable barrier, though it is generally assumed theyspawn downstream of the barrier if suitable habitat is available.Barrier-integrated traps remove some of these individuals prior tospawning, but like barriers, the efficacy of a trap for sea lampreycontrol depends on the fraction of the population removed (i.e.,exploitation rate) and the number of mature individuals that areleft to spawn. Primary challenges in evaluating a trap for sealamprey control are estimating the fraction of a population avail-able for capture at a trap location (i.e., vulnerability), the propor-tion of those that are caught (i.e., local trap efficiency), and thesize of the adult population. Understanding vulnerability and trapefficiency are critical to increasing exploitation because they al-low comparison of the potential benefits of improving existingtraps (to increase efficiency) versus adding traps in new locations(to increase vulnerability; Bravener and McLaughlin 2013).

Traps are also used to obtain mark–recapture estimates of adultabundance and exploitation rates; these estimates serve as the basisfor lake-wide adult population estimates (Mullett et al. 2003). Withintributaries, adult sea lamprey abundance has been estimated using amark–recapture method in which trap-caught lampreys are markedand released downstream for recapture in the same trap (or set oftraps). Abundance has been estimated using a modification ofSchaefer’s (1951) mark–recapture model (Mullett et al. 2003), whichrequires the assumption that marked and unmarked individuals areequally likely to be captured in the second sample (Pollock et al.1990). Potential violation of the “equal catchability” assumption is acritical uncertainty in trap operations because it could bias estimatesof trap efficiency and exploitation. Finally, these abundance esti-mates may not be indicative of entire river populations because themarked lampreys are released at a single location near each barrier.

Our goal was to evaluate a sea lamprey trap and barrier com-plex in the Cheboygan River (Fig. 1) using a multistate capture–recapture model with observations from acoustic telemetry andtraps. Recent advances in telemetry (Hockersmith and Beeman 2012;Cooke et al. 2013) and capture–recapture modeling (Perry et al. 2012;Royle et al. 2014) have allowed estimation of population-level fishmovement probabilities (e.g., passage, survival, collection, and spaceuse) while accounting for observation error (i.e., imperfect detection;Skalski et al. 2009; Buchanan and Skalski 2010; Perry et al. 2010).Although models of this class have been most commonly fit usingmaximum likelihood estimation (Perry et al. 2012), we used a Bayes-ian state-space model (BSSM) fit to the data using Markov chainMonte Carlo (MCMC) simulation (Gimenez et al. 2007; Calvert et al.

2009). Bayesian methods have recently become popular in ecology(see Buckland et al. 2004; King et al. 2010; Kéry and Schaub 2012)because they allow incorporation of information beyond the data(e.g., prior knowledge), facilitate estimation of the marginal distribu-tion of higher-level parameters in hierarchical models, and do notrequire asymptotic (i.e., large sample) assumptions (Brooks et al.2000; Ellison 2004; Kéry 2010). State-space models provide a flexibleand intuitive framework for building and fitting complex modelsbecause they explicitly separate ecological processes (e.g., move-ment, mortality) from sampling or observation processes (e.g., detec-tion, capture; Buckland et al. 2004; Gimenez et al. 2007; Calvert et al.2009). Specifically, our model used individual movement data de-rived from acoustic telemetry and trapping to estimate reach-specific fish movement probabilities (including dam passage andtrap capture probabilities) and to describe the spatial distributionand abundance of adult sea lampreys above and below the barrierduring riverine spawning migration.

The Cheboygan River, Michigan, a tributary to Lake Huron, hasconsistently experienced some of the largest catches of adult sealampreys in the Great Lakes. Despite a seemingly impassable barrier(consisting of a powerhouse, vessel passage lock, and separate spill-way) near the river mouth (Fig. 1A), the watershed above the lock anddam has remained infested with sea lampreys and has been treatedwith lampricide every 3 years at a cost ranging from $400 000 to$600 000 (USD). Although some evidence also exists that lakes in theupper Cheboygan River system support a landlocked population ofsea lampreys upstream of the lock and dam (Applegate 1950), it is notknown what fraction of the lower river population passes into theupper river, how many individuals escape above the dam, or whichroutes (e.g., vessel lock, turbine units, spill gates), if any, are used forupstream passage. Further, exploitation rate estimates for the Che-boygan River trap may be biased because marked lampreys are re-leased in the spill basin, so abundance estimates may not haveincluded lampreys that ceased migration farther downstream orpassed upstream through the powerhouse or lock before reachingthe spill basin.

To address these uncertainties, we estimated (i) the proportion ofthe population that passed the dam (i.e., upstream escapement rate);(ii) the number of individuals that passed the dam (i.e., escapement);(iii) the proportion of the lower river spawning population that wasvulnerable to the trap (i.e., proportion that reached the spill basinnear the trap); (iv) local trap efficiency (i.e., number of lampreycaught as a proportion of those present in the spill basin); and (v) theexploitation rate due to trapping. Our objectives addressed questionscritical for understanding effectiveness of barriers and traps to con-trol sea lamprey and for predicting benefits of control strategies thatare alternative to lampricides. The approach used in this study wasalso a novel application of BSSMs that may have broader applica-tions, including evaluation of barriers for other invasive species (e.g.,Asian carp (Hypophthalmichthys spp.)) and fish passage structures forother diadromous fishes (e.g., Pacific salmon (Oncorhynchus spp.),American eel (Anguilla rostrata)).

Materials and methods

Sea lamprey tagging and releaseAcoustic-tagged adult sea lampreys (N = 148) were released into the

lower Cheboygan River during spring 2011. Sea lampreys for tag im-plantation were collected from Carp Lake Outlet, a nearby tributaryto Lake Michigan, and were 405–576 mm in length (median 491 mm)and weighed 150–381 g (median 240 g). The sea lamprey trap in CarpLake Outlet was located about 500 m from the stream mouth, so sealampreys captured in the trap were early in their migration andusually not sexually mature. Acoustic tags (model V8-4L, Vemco, Hal-ifax, Nova Scotia, Canada) were 8 mm × 21 mm (diameter × length),weighed 2.0 g in air (0.9 g in water), had an expected minimum taglife of 84 days, and were programmed to transmit a uniquely en-coded signal with a power level of 146 dB (re 1 �Pa at 1 m) every

1714 Can. J. Fish. Aquat. Sci. Vol. 71, 2014

Published by NRC Research Press

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.

15–45 s for 4 days and then every 30–90 s for 80 days. Prior to surgery,each lamprey was anesthetized by immersion in 0.2 mL·L−1 clove oilsolution. Tags were surgically implanted through a 2 cm ventralincision near the midpoint between the posterior gill pore and theanterior dorsal fin. Each incision was closed with two interruptedsurgeon knots using a size 3-0 polydioxanone monofilament suture(PDSII, Ethicon), and glue (Vetbond, 3M) was applied to each knot.Sea lampreys were allowed to recover in an aerated tank for at least48 h prior to release. Every 1–2 days between 5 May and 13 June 2011,four tagged sea lampreys (two male, two female) were placed in aholding cage in the Cheboygan River about 1.3 km upstream fromLake Huron and about 1.2 km downstream of the lock and dam. Thecage door was opened at dusk for volitional release during the night.

Acoustic telemetry receiversA network of 21 autonomous acoustic telemetry receivers (VR2W,

Vemco) recorded a time-stamped movement history for each taggedfish during upstream migration (Fig. 1A). Receivers were placed atlocations to document approach to the lock and dam (five receiversat site 2A), presence near the trap (five receivers at site 3B), andescapement upstream of the dam (five receivers each at sites 3A and4A; one receiver at site 5A). Several receivers were placed at somesites for redundancy and for determination of routes used to pass thedam. Sentinel transmitters (model V13-1L and model V13T-1L; outputpower 147 and 150 dB, respectively; Vemco) were also placed in theriver above (sites 2A, 3B; V13-1L) and below the barrier (sites 3A, 4A;V13T-1L) to evaluate receiver performance.

Trap catch dataThe sea lamprey trap (site 4B; Fig. 1A) at the Cheboygan lock and

dam was operated once daily by the US Fish and Wildlife Service

between 17 April and 18 June 2011. The trap, a mesh box with twofunnel entrances leading to holding cages, was attached to the face ofthe dam below the spill gates so that attractant water flowed througheach holding cage. Each sea lamprey collected was scanned for pres-ence of an acoustic tag using a metal detector (R-8000 tunnel detec-tor, Northwest Marine Technology Inc., Shaw Island, Washington,USA) designed for detecting coded-wire tags and visually inspectedfor evidence of tagging (i.e., incision or sutures). All acoustic tagswere removed and implanted into another sea lamprey for releasedownstream. The elapsed time between release and recovery ranged1–35 days (median 9 days) among recovered lampreys, and all tagswere still pinging when recovered.

Data analysisParameter estimates were obtained with a three-step process.

First, a BSSM was used to estimate probabilities of movement amongriver reaches and into the trap while accounting for site-specificprobabilities of detection. Second, spatial distribution of the spawn-ing population (i.e., fraction of the population that reached eachsampling location) was estimated as a function of individual move-ment probabilities among reaches. Third, the number of untaggedlampreys caught in the trap was expanded by spatial distributionestimates to estimate abundance at each telemetry station and re-lease location.

Estimating movement probabilitiesThe BSSM included transition parameters that represented

probabilities of upstream movement among contiguous riverreaches (delineated by telemetry receivers) and trap capture(Fig. 1B). Each river channel was considered a separate state in themodel. For example, state A was assigned to the main river chan-

Fig. 1. Schematics of (A) the lower Cheboygan River (bottom inset shows general location of study site in the Great Lakes basin) withtelemetry receiver, sea lamprey release, and adult sea lamprey trapping sites; and (B) multistate mark–recapture model used to estimatetransition (�t�1,h,k) and detection (pt,k) probabilities for acoustic-tagged sea lamprey during 2011. Parameters in parentheses were not estimatedbecause of insufficient data.

Holbrook et al. 1715

Published by NRC Research Press

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.

nel that extended from the mouth through the vessel passage lockand into the upper river. State B was assigned to the route thatextended around the island toward the spillway outfall and trap,so a fish could only have entered the trap after moving fromstate A to state B. Although the trap could have been assigned aseparate state, we treated the trap as the last monitoring site instate B. Finally, state C represented cessation of upstream migra-tion (e.g., owing to death, spawning, or fallback) within a reach.

A tagged fish that entered any river reach could have ceasedmigration within that reach, continued migration upstream, orhave been captured in the trap. Each of those processes was esti-mated as a transition probability. Transition probabilities (�t,h,k)estimated the probability that a fish moved from state h at the tthupstream site (t = 1 at the release site and increments at eachtelemetry station or trap upstream) to state k at t + 1. In most cases,transition probabilities were considered movement probabilitiesand estimated the probability that a fish moved between any twosites. Alternatively, transition probabilities represented trap cap-ture probabilities (i.e., �3,B,B was the probability that a fish movedfrom the river at site 3B into the trap at site 4B) or dam passageprobabilities (i.e., �2,A,A was the probability that a fish moved fromsite 2A below the dam to site 3A above the dam).

Simply calculating the proportion of all fish that were detectedat each location would not provide unbiased estimates of fishdistribution, because a tagged fish may have passed a telemetryreceiver undetected. Therefore, the model included detectionprobabilities to prevent biased transition probability estimatescaused by observation error. Detection probabilities (pt,h) esti-mated the probability that a tagged fish was detected in state h atthe tth site, given that the fish reached that site.

The BSSM was a modification of models presented by Gimenezet al. (2007) and Calvert et al. (2009) and was fit to data usingMCMC (Gibbs sampling) with the software program WinBUGS(Spiegelhalter et al. 2003; Gimenez et al. 2009; Ntzoufras 2009;also see WinBUGS code in online Supplemental Material1) usingthe R package R2WinBUGS (Sturtz et al. 2005; R Development CoreTeam 2012). Input data were matrices wi,t and zi,t: wi,t were the datathat indicated that individual i was either observed (wi,t = 1)or not observed (wi,t = 0) at the tth site; zi,t was the partiallyobservable state matrix that provided the true state of individual iat the tth site. The true state was known where an individual wasobserved, but was inferred where an individual was not observed.For example, corresponding rows zi,· = [A A 0 B 0] and wi,· = [1 1 0 1 0]represented a fish that was released in state A, detected again instate A at t = 2 (i.e., first telemetry station upstream from release),not detected in any state at t = 3 (i.e., true state was unknown),caught in the trap at t = 4, and not detected at t = 5. Each row zidescribed the encounter history of a single lamprey, analogous tothe encounter histories (often called capture histories) of time-based capture–recapture models. However, the encounter histo-ries used in this model were spatial rather than temporal.

Posterior distributions of parameters were estimated by simu-lating draws from the joint posterior probability distribution:

(1) �(�, p|z, w) � �(z, w|�, p)�(�)�(p)

Thus, posterior probabilities (� (�, p | z, w)) of the parameters wereproportional to the product of the likelihood of the data given theparameters (�(z, w |�, p)) and the prior probabilities of movement(�(�)) and detection (�(p)). The median and mode of the posteriordistribution of each parameter were used as measures of centraltendency. Uncertainty was quantified using credible intervals cal-culated by determining the smallest interval for each parameterthat contained 95% of posterior samples. These highest posterior

density intervals (HPDI; Gelman et al. 2003; King et al. 2010) do notrequire distributional assumptions, incorporate all uncertaintyregarding information of other parameters, and were especiallyconvenient in this study because the shape of posterior distribu-tions varied among parameters. HPDI were calculated using theHPDinterval function in the R package coda (Plummer et al. 2006).

The full likelihood

(2) �(z, w|�, p) � �i�1

N �t�2

T[�(zi,t|zi,t�1, �t�1,j,k)�(wi,t|zi,t, pt,k)]

included a process likelihood (i.e., the probability of each state zi,tgiven the previous state zi,t–1 and the probability of transitioningbetween those states, �t�1,h,k) and an observation likelihood (i.e.,the probability that an individual was observed (wi,t) in state zi,tgiven that it was present at that site and that the detection prob-ability at that site was pt,k), where h = zi,t–1, k = zi,t, N was thenumber of tagged lamprey released, and T was the number ofencounter occasions, including release.

Process likelihoodThe true state zi,t was considered a categorical variable with

probabilities:

(3) �(zi,t � k|zi,t�1 � h) � ��t�1,h,k h � A, B; k � A, B, C0 h � C; k � A, B1 h � C; k � C

�For example, after release, a fish may have moved from state A tostate k with probability �1,A,k. Note that if a fish had ceased migration(h = C), it must have remained in the nonmigratory state (�t,C,C = 1), socould not later have been observed in any other state. Similarly, a fishcould not have been observed in another state after being caught inthe trap. The general form of the process likelihood (eq. 3) was fur-ther modified because some transitions could not have occurredgiven the structure of the study system. For example, �1,A,B = 0 be-cause a fish could not have moved directly from release to thetrapped state (Fig. 1B). Further, we assumed that upstream passagethrough spill gates was impossible (�3,B,A = 0). By applying theseconstraints, we assumed perfect prior knowledge about these pa-rameters. Consequently, four transition parameters were estimatedby the model. Only one transition (�2,A,k) remained multinomial be-cause the river split within that reach, allowing two possible migra-tory states (one for each pathway) and a third possible nonmigratorystate. Other transition parameters became binomial because theyrepresented movement through single, unbranched river reacheswith only two possible underlying states (i.e., migratory and nonmi-gratory). For binomial and multinomial transitions, flat beta andDirichlet prior probability distributions were used, respectively, be-cause no prior information was available about those parametersand because they are proper conjugate distributions for binomialand multinomial distributions (Gelman et al. 2003).

Observation likelihoodThe observation of each tagged fish at each site was considered

a Bernoulli random variable arising from the binominal detectionprobability pt,k and was conditional on presence at that location:

(4) �(wi,t � 1|zi,t � k) � �pt,k k � A, B0 k � C �

Therefore, each fish was only observable in states A and B andnever observable in state C. Further constraints were applied

1Supplementary data are available with the article through the journal Web site at http://nrcresearchpress.com/doi/suppl/10.1139/cjfas-2014-0581.

1716 Can. J. Fish. Aquat. Sci. Vol. 71, 2014

Published by NRC Research Press

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.

based on the structure of the study system and availability of data.For example, we set p1,B = 0 because site 1B did not exist and p4,B = 1because all acoustic-tagged fish were assumed to be identified bytrap personnel using a metal detector that was designed to detectmuch smaller coded-wire tags.

Detection probabilities were not estimable without some detec-tion information farther upstream. Therefore, under the generalmodel, detection and transition parameters were confounded atsites 4B and 5A, so only �4,B � �3,B,B × p4,B and �5,A � �4,A,A × p5,A,the joint probabilities of moving and being detected, were estima-ble at those locations. However, we were able to estimate �3,B,B(i.e., trap capture probability) because we assumed p4,B = 1. Similarly,no fish were detected above the dam (see Results), but we were ableto estimate �2,A,A (i.e., dam passage probability) because we assumedp3,A = 1 based on auxiliary data (i.e., sentinel tag detections). Afterthese constraints were applied, only two detection probabilities (p2,A

and p3,B) were estimable. Flat beta prior distributions (i.e., beta(1, 1))were assumed for p2,A and p3,B because no prior information wasavailable and because the beta distribution is a proper conjugatedistribution for binomial and Bernoulli distributions.

MCMC convergence diagnosticsInferences were based on 10 000 posterior samples for each

parameter. We discarded the first 10 000 samples (i.e., burn-in)from an initial chain of 110 000 and then retained every tenthsample (i.e., thinning) from the remaining chain. Burn-in wasdetermined from three short chains using Gelman and Rubin’s(1992) potential scale reduction factor (i.e., shrink factor). Totalchain length and thinning interval were determined using meth-ods described by Raftery and Lewis (1992). Convergence was con-firmed by examining autocorrelation, trace, and posterior densityplots for each parameter, including the deviance (Appendix A).

Assessing prior sensitivityTo determine if inferences were sensitive to assumptions of

perfect prior information about p3,A and p4,B, we compared resultsfrom the BSSM with results obtained using two alternative priorprobability distributions for each of those parameters (Appendix B).For each parameter, a “less informative” prior probability distribu-tion was chosen to represent hypothetical, but conservative, infor-mation that could have been obtained if auxiliary field tests had beenconducted during the study. A flat prior probability distribution wasalso chosen to determine how inferences would have changed if noprior information had existed about p3,A and p4,B.

Assessing model fitTo assess fit of the model to data, the observed frequency of

each unique encounter history was compared with the distribu-tion of frequencies predicted by the model. The predicted fre-quency of each encounter history was calculated from parameterestimates and compared with the corresponding observed fre-quency at each MCMC iteration. A posterior predictive P value wascalculated for each encounter history as the proportion of ex-pected frequencies that were greater than the observed fre-quency, plus one-half of the expected frequencies that were equalto the observed frequency. A predictive P value of 0.5 was consid-ered evidence that the parameter estimates provided by themodel were a perfect fit to the data, and fit was considered todecline as the P value approached 0 and 1.

Inferring the spatial distribution of the populationThe proportion of the stream spawning population that

reached (t,k) the tth monitoring site in state k was estimated as afunction of all possible transition probabilities that traced all pos-sible routes leading up to that site. Some spatial distribution pa-rameters were further defined in a management context (i.e.,vulnerability, exploitation, and escapement probabilities). Vul-nerability to the trap was defined as the fraction of the population

present in the spillway outfall at site 3B and was estimated as3,B � �1,A,A × �2,A,B. The exploitation rate was defined as the frac-tion of the population harvested in the trap at site 4B and wasestimated as 4,B � �1,A,A × �2,A,B × �3,B,B. The escapement rate wasdefined as the fraction of the population present above the barrierat site 3A and was estimated as 3,A � �1,A,A × �2,A,A. A posteriorprobability distribution was obtained for each derived parameterby calculating the value of each parameter for each MCMC itera-tion and then summarized as described for transition and detec-tion parameters.

Estimating abundance at release, receiver, and traplocations

Adult sea lamprey abundance was estimated at each samplelocation (i.e., release site and telemetry station) using a modifica-tion of the Peterson estimator (Seber 1982):

(5) Nt,k �ct,k

4,B

where t,k was the fraction of the population present at site tk,c is the number of unmarked lamprey caught in the trap, and4,B was the fraction of the population caught in the trap. Allpotential spawners were assumed to have passed the release site(i.e., 1,A = 1), because no spawning habitat was known betweenthe river mouth and release site. Therefore, N1,A was defined as thetotal population size and N3,A as the number of sea lampreys thatpassed upstream of the barrier (i.e., escapement).

Model assumptionsModel assumptions included the following: (1) All tags were cor-

rectly identified, no tags were lost, and no tags failed during thestudy period. (2) Tagged and untagged lampreys had equal transition(i.e., movement, dam passage) and trap capture probabilities. (3) Ateach sampling location, every tagged lamprey had the same proba-bility of detection and transition. (4) For each tagged lamprey, detec-tion and transition probabilities were independent among samplinglocations. Violation of assumption 1 would result in biased transitionprobability estimates because incorrect tag identification, tag loss, orpremature tag failure (e.g., battery failure during the study) would beincorrectly interpreted as cessation of upstream migration. We as-sumed that such violations were rare because all recovered tags werefunctional and because no tag loss was observed from recoveries ofsea lampreys that were double-tagged with acoustic and coded-wiretags (C.M. Holbrook, unpublished data). Assumption 2 could havebeen violated if tagged lampreys were not a representative sample ofthe untagged population in this study or if fate of tagged fish wasaffected by tagging. For our study, lampreys were purposely collectedfrom a source outside of the study stream so that tagged fish were“naïve” to the study system to reduce the likelihood of bias caused byprevious experience or capture heterogeneity (Pollock et al 1990). Useof animals captured in other streams should not have affected theirmigration patterns because sea lampreys do not home to natalstreams (Bergstedt and Seelye 1995). Although we did not expecttagging to affect behavior, timing of our releases may not havematched timing of migration through the lower Cheboygan River.The implications of this assumption were further explored by testingthe hypothesis that trap capture probability was related to releasedate. Assumptions 3 and 4 could have been violated if detection,movement, or trap capture probabilities varied among individuals(e.g., owing to differences in sex, mass, or maturity) or over time (e.g.,owing to changes in discharge or water temperature). Violations ofassumptions 3 and 4 were expected to be evident as lack of fit inposterior predictive checks (see section on Assessing model fit).

Holbrook et al. 1717

Published by NRC Research Press

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.

Testing the assumption of constant capture probabilityTo test the hypothesis that the probability of trap capture was

constant during the trapping season, we modified the likelihood todefine �3,B,B as a function of release date. Specifically, the logit linkfunction (logistic regression) was used to estimate logit(�3,B,B,i) as alinear combination of parameters 1 (intercept), 2 (slope), and re-lease date (e.g., logit(�3,B,B,i) = 1 + 2 × xi), where xi and �3,B,B,i repre-sented release date and trap capture probability, respectively, for theith tagged fish. We used a normal distribution with mean of 0 andvariance of 2.72 (i.e., precision = 0.368) as prior probability distribu-tions for 1 and 2 because that distribution is essentially flat on thelogit scale (see Lunn et al. 2012). Posterior samples were obtainedusing MCMC simulation as described above. Estimates of �3,B,B for aspecific release date were back-calculated using the inverse logitfunction. To test the hypothesis that �3,B,B was constant over time, wecalculated the proportion of posterior samples where 2 < 0 (i.e.,decreasing over time) and, separately, where 2 > 0 (i.e., increasingover time).

Results

Assessing model fitAll observed encounter history frequencies were contained in

95% HPDI for expected frequencies (Fig. 2). Observed and expected(posterior median) frequencies differed by less than three fishamong all encounter histories and one fish for the most “poorlypredicted” encounter history (Table 1).

Movement and detection probabilitiesIn the lower river, 144 of 148 tagged sea lampreys released were

detected at telemetry receivers, but no tagged sea lampreys weredetected at telemetry stations above the dam. Although detectionprobabilities could not be estimated upstream of the dam, sentineltag detections indicated that conditions were more favorable fordetection at sites upstream of the dam (sites 3A and 4A) than belowthe dam, where detection probability estimates ranged 88%–98%

(Table 2). Of adult sea lampreys that entered the river, 94%–100%moved upstream to the powerhouse (�1,A,A; Table 2; Fig. 3). From thepowerhouse tailrace, 0%–2% of the population passed the dam intothe upper river (�2,A,A) and 86%–98% approached the spillway outfallcontaining the trap (�2,A,B). Of sea lampreys that were present in thespillway outfall, 52%–69% were caught in the trap (�3,B,B).

Spatial distribution and abundanceThe trap caught 13 580 unmarked and 81 acoustic-tagged sea lam-

preys. Based on transition probability estimates, 83%–96% (Table 3) ofthe spawning population reached the trap site (i.e., vulnerability,3,B), 46%–61% were caught in the trap (i.e., exploitation, 4,B), and0%–2% passed upstream through the lock and dam facility (i.e., es-capement rate, 3,A). Based on spatial distribution estimates, 21 828

Table 1. Observed encounter history frequenciesfrom detections of tagged sea lampreys in theCheboygan River during 2011, with expected fre-quencies (posterior median) predicted from theBayesian state-space model and the proportion ofposterior expected frequencies that were more ex-treme than the observed frequency (P value).

Encounterhistory

Observedfrequency

Expectedfrequency P value

A0000 4 5 0.625A00B0 1 0 0.025A0A00 0 0 0.500A0B00 0 1 0.914A0BB0 1 1 0.708AA000 16 16 0.535AA0B0 8 9 0.651AAA00 0 1 0.794AAB00 47 45 0.355AABB0 71 68 0.312

Table 2. Transition and detection probability estimates (posteriormode and median) with 95% highest posterior density intervals (HPDI)for sea lampreys in the Cheboygan River during 2011 (see Fig. 1).

ParameterPosteriormode

Posteriormedian 95% HPDI

�1,A,A 0.978 0.971 0.940–0.995

�1,A,C � 1 � �1,A,A 0.022 0.029 0.008–0.065

�2,A,A 0.001 0.005 0.000–0.020

�2,A,B 0.919 0.920 0.857–0.982

�2,A,C � 1 � �2,A,A � �2,A,B 0.073 0.073 0.013–0.135

�3,B,B 0.602 0.602 0.516–0.685

�3,B,C � 1 � �3,B,B 0.398 0.398 0.316–0.486

p2,A 0.983 0.979 0.951–0.998

p3,B 0.884 0.883 0.811–0.944

Table 3. Estimates (posterior median with 95% highestposterior density intervals (HPDI) in parentheses) of adultsea lamprey abundance (Nt,k) and proportion of the pop-ulation (t,k) that reached each acoustic monitoring sitein the Cheboygan River during 2011 (see Fig. 1).

Site (tk) t,k Nt,k

1A 1.000a 24 958 (21 828, 29 299)2A 0.978 (0.940–0.995) 24 331 (21 250, 28 375)3A 0.001 (0.000–0.020) 34 (0, 514)3B 0.891 (0.826–0.956) 22 257 (19 698, 26 166)4B 0.542 (0.457–0.613) 13 580b

aNo HPDI is available because we assumed 1,A = 1 (i.e., allindividuals in the population passed the release site).

bN4,B is the number of unmarked sea lampreys caught in the trap.

Fig. 2. Observed frequencies of unique encounter histories againstexpected frequencies (posterior median) under the Bayesian state-space model. 95% CRI = 95% credible intervals.

1718 Can. J. Fish. Aquat. Sci. Vol. 71, 2014

Published by NRC Research Press

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.

Fig. 3. Posterior distributions of transition parameters (see Fig. 1) representing movement, dam passage, and trap capture probabilities foracoustic-tagged sea lampreys in the lower Cheboygan River.

Fig. 4. Posterior distributions of derived parameters representing the estimated population proportions and abundances of sea lampreys atsites 2A (top panels); 3A (middle panels); and 3B (bottom panels) in the lower Cheboygan River (Fig. 1).

Holbrook et al. 1719

Published by NRC Research Press

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.

to 29 300 sea lampreys (95% HPDI) entered the river from Lake Huron(i.e., passed the release site; N1,A) and 0 to 514 passed above the dam(N3,A) during 2011. Posterior distributions for derived parameters de-viated from normality, especially near the boundaries of the bino-mial distribution (Fig. 4).

Testing the assumption of constant capture probabilityTrap capture probability decreased during the trapping season

(Fig. 5). We estimated (95% HPDI) that 1 ranged from 0.249 to1.734 (median = 0.979), 2 ranged from –0.061 to –0.004 (median =–0.029), 96.0% of 2 posterior samples were less than zero, and4.0% of 2 posterior samples were greater than zero. Back-calculatedestimates of the trap capture probability (�3,B,B) ranged from 72%(posterior median) for lampreys released at the beginning of thestudy on 6 May to 46% at the end of the study on 13 June.

Discussion

Implications for sea lamprey control and assessment

Evaluating the lock and dam as a sea lamprey barrierAlthough the Cheboygan Dam allowed no tagged fish to pass

above the barrier, we estimated (based on our sample of 148 taggedlampreys in a population of about 25 000 lampreys) that 0%–2% ofthe untagged population could have escaped upstream above thedam without passage of a single acoustic-tagged lamprey. Al-though data from this study did not reveal any upstream passageroutes at the barrier, we concluded, after further inspection of thepowerhouse, spillway, and lock, that the vessel lock would haveprovided the most plausible route of upstream passage.

To ensure low recruitment in sea lamprey populations, Dawsonand Jones (2009) recommended that sea lamprey control agentsshould aim for stream-specific spawner abundances less than0.2 females per 100 m2 of larval habitat. Applying that criterion toroughly 425 000 m2 of preferred larval habitat upstream of theCheboygan Dam (A. Jubar, US Fish and Wildlife Service, Luding-ton, Michigan, personal communication, 2013) yields a targetthreshold of 1700 adult sea lampreys (assuming equal sex ratio).Our best inference (with 95% probability) was that the number of

sea lampreys passing the dam was between 0 and 514, so we con-clude that the Cheboygan Dam was an effective sea lamprey bar-rier under the criterion proposed by Dawson and Jones (2009).Alternatively, an effective sea lamprey barrier has been defined bycontrol agents as one that prevents or delays upstream lampricidetreatments by limiting passage to a sufficiently small number ofindividuals. We do not know precisely how many sea lampreyswould be necessary to colonize the upper river to produce suffi-cient recruitment to trigger a lampricide treatment, but we rea-son that our sample size was insufficient to adequately evaluatebarrier effectiveness for large populations as in the CheboyganRiver. For example, we would have needed to release more than1643 tagged lampreys to conclude (with 95% probability) thatfewer than 50 tagged lampreys could have passed above the damwhen no passage of tagged fish was observed.

Although the estimated sea lamprey population escaping up-stream of the dam was less than 514, the upper river has requiredregular lampricide treatment since the 1960s, and sea lampreyreproduction and recruitment was observed above the dam dur-ing the year of our investigation (J. Slade, US Fish and WildlifeService, Ludington, Michigan, personal communication, 2013). Ev-idently, sea lamprey are not prone to recruitment failures evenwhen population density is low, consistent with their history ofrapid colonization of the Great Lakes (Smith and Tibbles 1980).While the presence of a landlocked sea lamprey population abovethe dam has been supported by accounts from anglers of parasiticlamprey in upstream lakes of the Cheboygan River system andby Applegate (1950), results from this study do not rule out thepossibility that spawners from Lake Huron have contributed torecruitment in the upper river. Prior to taking expensive manage-ment actions to eliminate possible sea lamprey passage at the lockand dam, confirmation of lamprey passage at the lock and addi-tional information on sea lamprey ecology and life history in theupper river should be obtained.

Evaluating the sea lamprey trap: vulnerability and exploitationOur results suggest that improved trapping at the current loca-

tion has greater potential to increase exploitation (46%–61% wereremoved) than installation of traps at new locations, because 83%–96% of the population (3,B) was vulnerable to existing traps (be-cause they used route B and approached the spillway) while only52%–69% of those were caught (�3,B,B). Although results suggestthat catch is limited more by trap performance than trap location,previous mark–recapture studies have shown that the CheboyganRiver trap is one of the most efficient in the Great Lakes. Goodperformance has been attributed to the trap being located in asmall basin with circular flow pattern below an impassable spill-way. Recent studies of sea lamprey behavior around similar trapsin the St. Marys River have suggested that many lampreys remainclose to the river bottom near traps and do not encounter en-trances to surface-oriented traps (R.L. McLaughlin, personal com-munication). Future studies should aim to determine if similardynamics are limiting efficiency of traps in the Cheboygan River.

Implications of decreasing capture probability on populationassessment

Decreasing trap capture probability during the trapping season(Fig. 5) supports the use of a time-stratified model for adult assess-ment because lampreys that were released earlier in the seasonwere more likely to be captured than lampreys released later inthe season. Interestingly, the nonstratified BSSM and the modi-fied Schaeffer model yielded similar point estimates of trapcapture probability (BSSM: 0.60; modified Schaeffer: 0.62) andabundance near the trap site (BSSM: 22 257; modified Schaeffer:21 986), which suggested that they may have been affected simi-larly by temporal variation in capture probability. Although ourestimate of trap capture probability when �3,B,B was assumed con-stant (52%–69%) was close to the capture probability for sea lam-

Fig. 5. Estimated trap capture probability as a function of release datefor acoustic-tagged sea lamprey in the Cheboygan River, 2011. The solidline shows fitted values based on posterior median fitted values, andbroken lines represent 95% credible intervals. Ticks (with randomhorizontal jitter to prevent overlap; <0.5 day) show the distribution ofthe data (0 = never caught in trap; 1 = caught in trap).

1720 Can. J. Fish. Aquat. Sci. Vol. 71, 2014

Published by NRC Research Press

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.

prey released during the middle of the trapping season (Fig. 5),more accurate estimation of population spatial distribution andabundance may require further assumptions about the timing ofriver entry. For example, if more sea lampreys entered the riverearly in the season than late in the season, then greater weightwould need to be applied to encounter histories of early-releasedfish. Although declining capture probability may confirm theneed for a time-stratified model, it also supports concern that useof a single capture method at a single location may violate theassumption of equal catchability in the abundance estimator.Future research should be directed toward understanding thetiming of entry into streams and the relation between trap cap-ture probability and environmental covariates. Such informationcould guide the development of improved assessment methods.

Advantages, limitations, and challenges of the Bayesianstate-space approach

State-space approaches are conceptually well-suited to estima-tion of reach-specific transition and site-specific detection proba-bilities from telemetry data because of the conditional nature offish movement through river systems and the potential for imper-fect detection at any site. Indeed, the goodness of fit test showedthat observed encounter histories were not improbable under themodel (Fig. 2), suggesting that any violations of assumptions 3 and4 were small. For example, if tagged lampreys that migrated dur-ing high river discharge were less likely to be detected at sites 2Aand 3B than lampreys migrating during lower discharge, wemight have observed more encounter histories like “A00B0” andfewer encounter histories like “AA0B0” and “A0BB0” than ex-pected under the model. Similar processes likely contributed tominor deviations from perfect fit (Fig. 2), as expected for anymodel.

Quantifying uncertaintyAccurate accounting of uncertainty is a basic element of infer-

ence (Royle and Dorazio 2008) and can be especially important intelemetry studies because small sample sizes are common (causedby the high cost of tags) and because telemetry data are often usedto evaluate structures or strategies that are considered critical forinvasive species control (in this case, a barrier for sea lampreycontrol) or native species conservation or restoration. In manyapplications, individual parameters of a mark–recapture modelare of less interest or use than parameters that are estimated ascombinations of individual model parameters. In Bayesian mod-els, derived parameters can be calculated for each MCMC iterationto yield a posterior distribution for each parameter that can besummarized using percentiles or HPDIs. In our experience, cred-ible interval construction is easy with MCMC and more intuitivethan analogous methods for maximum-likelihood estimates (e.g.,Delta method; Seber 1982) that rely on asymptotic (e.g., large sam-ple) assumptions.

Accurate estimation of uncertainty can be challenging when anestimate is on the boundary of the binomial distribution (i.e., nosuccesses or failures were observed). This situation has receivedmuch more attention in medicine (e.g., for estimating risk ofcomplications after a procedure when none have been observed;Hanley and Lippman-Hand 1983; Eypasch et al. 1995) than in ecol-ogy. In such cases, confidence intervals can be approximated withan “exact confidence interval” (Clopper and Pearson 1934) or bythe simple “rule of three” (i.e., 3/n, where n is the number of trials;Hanley and Lippman-Hand 1983), but these methods do not allowuncertainty to be included in the model. Bayesian models canaccurately characterize uncertainty in these instances by usinginformation from the prior probability distribution in addition tothe likelihood function. For example, the BSSM produced a poste-rior distribution for dam passage with a nominal estimate (poste-rior mode) matching the observed proportion (e.g., �2,A,A = 0.00)and a 95% HPDI (0.00–0.02) representing uncertainty due to our

sample size. These results are similar to those that could havebeen obtained with profile likelihood confidence intervals if wehad fit our model using maximum likelihood estimation.

Sensitivity to prior distributionsPrior probability distributions were used for three purposes in

this study. First, we used priors to apply structural constraintswhen transition or detection was physically impossible. Second,“flat” priors were used when we had no prior information aboutthe value of a parameter (i.e., all values within the parameterspace were equally plausible a priori). Use of a flat prior is not freeof criticism, because it implies that all values within the parame-ter space were equally plausible a priori. This was our choice,however, because of lack of better information. Third, we usedpriors to specify assumptions needed to estimate parameters thatwere confounded with other parameters. Based on conservativeestimates, inferences would not have changed if we had used “lessinformative” priors, such as those that might have been obtainedfrom auxiliary field tests, for the probabilities of detection in thetrap and upstream of the dam (Appendix B). Use of a flat prior forthe probability of detection upstream of the dam would haveincreased uncertainty about the proportion of the population thatpassed the dam and the number of individuals that escaped up-stream, but would not have affected any other parameter esti-mate. Use of a flat prior for probability of detection in the trapwould have produced a larger estimate of trap efficiency andsmaller estimate of abundance, but the trend still would havesuggested, albeit with greater uncertainty, that catch was limitedby trap efficiency more than location. Although Bayesian modelsare often criticized for use of subjective prior distributions (Efron1986; Dennis 1996), we view prior distributions as valuable toolsthat provide formal mechanisms to specify and test assumptions.

Despite many perceived advantages, several limitations andchallenges may prevent widespread use of BSSMs for fish move-ment analysis. First, more user-friendly software applications maybe needed for widespread adoption of Bayesian approaches. Therecent addition of Bayesian estimation to the popular softwareprogram MARK (White et al. 2006) is one example. Second, we arenot aware of any widely accepted model selection procedures forBSSMs (see Kéry and Schaub 2012), although one might emergefrom recent progress on Bayesian variable selection methods(O’Hara and Sillanpää 2009; King et al. 2010). Finally, although theunidirectional (e.g., upstream-only) structure of the model in thisstudy was useful for estimating upstream passage probabilities, amodel that allows for movement in both directions could be use-ful or needed in future fish passage evaluations. For example, ifany lampreys had passed upstream of the dam, then we wouldhave desired an estimate of the proportion of those that later “fellback” over the dam. The need for such models may not be limitedto measuring fallback over dams (see Frank et al. 2009), but couldalso be used to identify locations where migrants show reversalsor other behavioral changes that could increase exposure to pred-ators or other risks. To some degree, we suspect that the tendencyto use unidirectional models in fish passage studies (Skalski et al.2009; Perry et al. 2010; Holbrook et al. 2011) can be attributed tothe historical use of capture–recapture models to estimate demo-graphic parameters through time (Arnason 1972; Seber 1982;Lebreton et al. 1992; Brownie et al. 1993) rather than space. Aprimary difference between time-based models and space-basedmodels is that movement through time is unidirectional, andtherefore conditional, but movement through space may be un-constrained. A primary challenge in developing multistate capture–recapture models that allow for reversals is state-specific estimationof detection probabilities. Promising new approaches have re-cently emerged (Buchanan and Skalski 2010; Raabe et al. 2014), butthere remains a need to better define and distinguish spatialcapture–recapture models (in this case discrete-space) from theirtime-based analogs (see Royle et al. 2014).

Holbrook et al. 1721

Published by NRC Research Press

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.

AcknowledgementsThis work was funded by the Great Lakes Fishery Commi-

ssion through Great Lakes Restoration Initiative appropriations(GL-00E23010-3). The Great Lakes Acoustic Telemetry Observa-tion System (http://www.data.glos.us/glatos) assisted with project co-ordination. Great Lakes Tissue and the Michigan Department ofNatural Resources provided property access. T. Brenden, M. Hansen,and two anonymous reviewers provided constructive sugges-tions to improve the manuscript. J. Barber, E. Benzer, E. Larson,H. Thompson, Z. Holmes, and M. Lancewicz provided assistance.Any use of trade, product, or firm names is for descriptive pur-poses only and does not imply endorsement by the US Govern-ment. The findings and conclusions in this article are those of theauthors and do not necessarily represent the views of the US Fishand Wildlife Service. This article is contribution No. 4 of the GreatLakes Acoustic Telemetry Observation System, contribution No. 1862of the USGS Great Lakes Science Center, and contribution No. 2014-11of the MSU Quantitative Fisheries Center.

ReferencesApplegate, V.C. 1950. Natural history of the sea lamprey, Petromyzon marinus, in

Michigan. US Fish and Wildlife Service Special Scientific Report Fisheries 55.Arnason, A.N. 1972. Parameter estimates from mark-recapture experiments on

two populations subject to migration and death. Res. Popul. Ecol. 13: 97–113.doi:10.1007/BF02521971.

Bergstedt, R.A., and Seelye, J.G. 1995. Evidence for lack of homing by sea lam-preys. Trans. Am. Fish. Soc. 124: 235–239. doi:10.1577/1548-8659(1995)124<0235:EFLOHB>2.3.CO;2.

Bravener, G.A., and McLaughlin, R.L. 2013. A behavioural framework for trap-ping success and its application to invasive sea lamprey. Can. J. Fish. Aquat.Sci. 70(10): 1438–1446. doi:10.1139/cjfas-2012-0473.

Brooks, S.P., Catchpole, E.A., and Morgan, B.J.T. 2000. Bayesian animal survivalestimation. Stat. Sci. 15: 357–376. doi:10.1214/ss/1009213003.

Brownie, C., Hines, J.E., Nichols, J.D., Pollock, K.H., and Hestbeck, J.B. 1993.Capture–recapture studies for multiple strata including non-Markovian tran-sitions. Biometrics, 49: 1173–1187. doi:10.2307/2532259.

Buchanan, R.A., and Skalski, J.R. 2010. Using multistate mark–recapture meth-ods to model adult salmonid migration in an industrialized river. Ecol.Model. 221: 582–589. doi:10.1016/j.ecolmodel.2009.11.014.

Buckland, S.T., Newman, K.B., Thomas, L., and Koesters, N.B. 2004. State-spacemodels for the dynamics of wild animal populations. Ecol. Modell. 171: 157–175. doi:10.1016/j.ecolmodel.2003.08.002.

Calvert, A.M., Bonner, S.J., Jonsen, I.D., Flemming, J.M., Walde, S.J., andTaylor, P.D. 2009. A hierarchical Bayesian approach to multi-state mark–recapture: simulations and applications. J. Appl. Ecol. 46: 610–620. doi:10.1111/j.1365-2664.2009.01636.x.

Christie, G.C., and Goddard, C.I. 2003. Sea Lamprey International Symposium(SLIS II): advances in the integrated management of sea lamprey in the GreatLakes. J. Gt. Lakes Res. 29: 1–14. doi:10.1016/S0380-1330(03)70474-2.

Clopper, C., and Pearson, E.S. 1934. The use of confidence or fiducial limitsillustrated in the case of the binomial. Biometrika, 26: 404–413. doi:10.1093/biomet/26.4.404.

Cooke, S.J., Midwood, J.D., Thiem, J.D., Klimley, P., Lucas, M.C., Thorstad, E.B.,Eiler, J., Holbrook, C., and Ebner, B.C. 2013. Tracking animals in freshwaterwith electronic tags: past, present and future. Animal Biotelemetry, 1: 5.doi:10.1186/2050-3385-1-5.

Dawson, H.A., and Jones, M.L. 2009. Factors affecting recruitment dynamics ofGreat Lakes sea lamprey (Petromyzon marinus) populations. J. Gt. Lakes Res. 35:353–360. doi:10.1016/j.jglr.2009.03.003.

Dennis, B. 1996. Discussion: should ecologists become Bayesians? Ecol. Appl. 6:1095–1103. doi:10.2307/2269594.

Efron, B. 1986. Why isn’t everyone a Bayesian? Am. Stat. 40: 1–5. doi:10.2307/2683105.

Ellison, A.M. 2004. Bayesian inference in ecology. Ecol. Lett. 7: 509–520. doi:10.1111/j.1461-0248.2004.00603.x.

Eypasch, E., Lefering, R., Kum, C.K., and Troidl, H. 1995. Probability of adverseevents that have not yet occurred: a statistical reminder. BMJ, 311: 619–620.doi:10.1136/bmj.311.7005.619.

Frank, H.J., Mather, M.E., Smith, J.M., Muth, R.M., Finn, J.T., and McCormick, S.D.2009. What is “fallback”?: metrics needed to assess telemetry tag effects onanadromousfishbehavior.Hydrobiologia,635(1):237–249.doi:10.1007/s10750-009-9917-3.

Gelman, A., and Rubin, D.B. 1992. Inference from iterative simulation usingmultiple sequences. Stat. Sci. 7: 457–472. doi:10.1214/ss/1177011136.

Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. 2003. Bayesian data analysis.2nd ed. Chapman and Hall, Boca Raton, Fla.

Gimenez, O., Rossi, V., Choquet, R., Dehais, C., Doris, B., Varella, H., Vila, J.-P.,

and Pradel, R. 2007. State-space modelling of data on marked individuals.Ecol. Model. 206: 431–438. doi:10.1016/j.ecolmodel.2007.03.040.

Gimenez, O., Bonner, S., King, R., Parker, R.A., Brooks, S., Jamieson, L.E.,Grosbois, V., Morgan, B.J.T., and Thomas, L. 2009. WinBUGS for populationecologists: Bayesian modeling using Markov Chain Monte Carlo methods. InModeling demographic processes in marked populations. Edited byD.L. Thomson, E.G. Cooch, and M.J. Conroy. Springer Series: EnvironmentalEcological Statistics, Vol. 3. pp. 883–916.

Hanley, J.A., and Lippman-Hand, A. 1983. If nothing goes wrong, is everything allright? J. Am. Med. Assoc. 249: 1743–1745. doi:10.1001/jama.1983.03330370053031.

Hansen, M.J. 1999. Lake trout in the Great Lakes: basin-wide stock collapse andbinational restoration. In Great Lakes fishery policy and management: abinational perspective. Edited by W.W. Taylor and C.P. Ferreri. Michigan StateUniversity Press, East Lansing, Mich. pp. 417–453.

Hockersmith, E.E., and Beeman, J.W. 2012. A history of telemetry in fisheryresearch. In Telemetry techniques: a user guide for fisheries research. Editedby N.S. Adams, J.W. Beeman, and J.H. Eiler. American Fisheries Society,Bethesda, Md. pp. 7–19.

Holbrook, C.M., Zydlewski, J., and Kinnison, M.T. 2011. Survival of migratingAtlantic salmon smolts through the Penobscot River, Maine, U.S.A.: a pre-restoration assessment. Trans. Am. Fish. Soc. 140: 1255–1268. doi:10.1080/00028487.2011.618356.

Johnson, N.S., Thompson, H.T., Holbrook, C., and Tix, J.A. 2014. Blocking andguiding adult sea lamprey with pulsed direct current from vertical elec-trodes. Fish. Res. 150: 38–48. doi:10.1016/j.fishres.2013.10.006.

Jones, M.L. 2007. Toward improved assessment of sea lamprey population dy-namics in support of cost-effective sea lamprey management. J. Gt. Lakes Res.33: 35–47. doi:10.3394/0380-1330(2007)33[35:TIAOSL]2.0.CO;2.

Kéry, M. 2010. Introduction to WinBUGS for ecologists. A bayesian approach toregression, ANOVA, mixed models and related analyses. Academic Press,Burlington, Mass.

Kéry, M., and Schaub, M. 2012. Bayesian population analysis using WinBUGS: ahierarchical perspective. Academic Press, Burlington, Mass.

King, R., Morgan, B.J.T., Gimenez, O., and Brooks, S.P. 2010. Bayesian analysis forpopulation ecology. Chapman and Hall, Boca Raton, Fla.

Lavis, D.S., Hallet, A., Koon, E.M., and McAuley, T.C. 2003. History of and ad-vances in barriers as an alternative method to suppress sea lampreys in theGreat Lakes. J. Gt. Lakes Res. 29: 362–372. doi:10.1016/S0380-1330(03)70500-0.

Lebreton, J.D., Burnham, K.P., Clobert, J., and Anderson, D.R. 1992. Modelingsurvival and testing biological hypotheses using marked animals: a unifiedapproach with case studies. Ecol. Monogr. 62: 67–118. doi:10.2307/2937171.

Lunn, D., Jackson, C., Best, N., Thomas, A., and Spiegelhalter, D. 2012. The BUGSbook: a practical introduction to Bayesian analysis. CRC Press, Boca Raton, Fla.

McLaughlin, R.L., Hallett, A., Pratt, T.C., O’Connor, L.M., and McDonald, D.G.2007. Research to guide use of barriers, traps, and fishways to control sealamprey. J. Gt. Lakes Res. 33: 7–19. doi:10.3394/0380-1330(2007)33[7:RTGUOB]2.0.CO;2.

Mullett, K.M., Heinrich, J.W., Adams, J.V., Young, R.J., Henson, M.P.,McDonald, R.B., and Fodale, M.F. 2003. Estimating lake-wide abundance ofspawning-phase sea lampreys (Petromyzon marinus) in the Great Lakes: extrap-olating from sampled streams using regression models. J. Gt. Lakes Res. 29:240–252. doi:10.1016/S0380-1330(03)70492-4.

Ntzoufras, I. 2009. Bayesian modeling using WinBUGS. Wiley, Hoboken, N.J.O’Hara, R.B., and Sillanpää, M.J. 2009. A review of Bayesian variable selection

methods: what, how and which. Bayesian Anal. 4(1): 85–117. doi:10.1214/09-BA403.

Perry, R.W., Skalski, J.R., Brandes, P.L., Sandstrom, P.T., Klimley, A.P.,Ammann, A., and MacFarlane, B. 2010. Estimating survival and migrationroute probabilities of juvenile Chinook salmon in the Sacramento–SanJoaquin River Delta. N. Am. J. Fish. Manage. 30: 142–156. doi:10.1577/M08-200.1.

Perry, R.W., Castro-Santos, T., Holbrook, C.M., and Sanford, B. 2012. Usingmark–recapture models to estimate survival from telemetry data. In Telemetrytechniques: a user guide for fisheries research. Edited by N.S. Adams, J.W. Bee-man, and J.H. Eiler. American Fisheries Society, Bethesda, Md. pp. 453–475.

Plummer, M., Best, N., Cowles, K., and Vines, K. 2006. CODA: convergence diag-nosis and output analysis for MCMC. RNews, 6: 7–11.

Pollock, K.H., Nichols, J.D., Brownie, C., and Hines, J.E. 1990. Statistical inferencefor capture–recapture experiments. Wildlife Monogr. 107: 1–97.

R Development Core Team. 2012. R: a language and environment for statisticalcomputing. R Foundation for Statistical Computing, Vienna, Austria.

Raabe, J.K., Gardner, B., and Hightower, J.E. 2014. A spatial capture–recapturemodel to estimate fish survival and location from linear continuous moni-toring arrays. Can. J. Fish. Aquat. Sci. 71(1): 120–130. doi:10.1139/cjfas-2013-0198.

Raftery, A.E., and Lewis, S.M. 1992. One long run with diagnostics: Implementa-tion strategies for Markov chain Monte Carlo. Stat. Sci. 7: 493–497. doi:10.1214/ss/1177011143.

Royle, J.A., and Dorazio, R.M. 2008. Hierarchical modeling and inference inecology. The analysis of data from populations, metapopulations and com-munities. Academic Press, New York.

Royle, J.A., Chandler, R.B., Sollmann, R., and Gardner, B. 2014. Spatialcapture–recapture. Academic Press, Waltham, Mass.

1722 Can. J. Fish. Aquat. Sci. Vol. 71, 2014

Published by NRC Research Press

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.

Schaefer, M.B. 1951. Estimation of size of animal populations by marking exper-iments. Fish. Bull. 52: 187–203.

Seber, G.A.F. 1982. The estimation of animal abundance and related parameters.Macmillan, New York.

Skalski, J.R., Buchanan, R.A., Townsend, R.L., Steig, T.W., and Hemstrom, S.2009. A multiple-release model to estimate route-specific and dam passagesurvival at a hydroelectric project. N. Am. J. Fish. Manage. 29: 670–679. doi:10.1577/M07-227.1.

Smith, B.R., and Tibbles, J.J. 1980. Sea lamprey (Petromyzon marinus) in LakesHuron, Michigan, and Superior: history of invasion and control, 1936–78.Can. J. Fish. Aquat. Sci. 37(11): 1780–1801. doi:10.1139/f80-222.

Spiegelhalter, D.J., Thomas, A., Best, N.G., and Lunn, D. 2003. WinBUGS usermanual. Version 1.4. MRC Biostatistics Unit, Cambridge, UK.

Sturtz, S., Ligges, U., and Gelman, A. 2005. R2WinBUGS: a package for runningWinBUGS from R. J. Stat. Softw. 12: 1–16.

Swink, W.D. 1999. Effectiveness of an electrical barrier in blocking a sea lampreyspawning migration on the Jordan River, Michigan. N. Am. J. Fish. Manage.19: 397–405. doi:10.1577/1548-8675(1999)019<0397:EOAEBI>2.0.CO;2.

White, G.C., Kendall, W.L., and Barker, R.J. 2006. Multistate survival models andtheir extensions in Program MARK. J. Wildl. Manage. 70: 1521–1529. doi:10.2193/0022-541X(2006)70[1521:MSMATE]2.0.CO;2.

Appendix AThis appendix describes methods used to determine burn-in,

chain length, thinning interval, and confirm convergence for theBayesian state-space model used to estimate movement, trap cap-ture, dam passage, and detection probabilities of sea lampreys inthe Cheboygan River during 2011.

We used the potential scale reduction factor (i.e., shrink factor;Gelman and Rubin 1992; Brooks and Gelman 1998) as criteria forthe number of samples to discard at the start of each MCMC chain(i.e., burn-in). The minimum number of burn-in samples was esti-mated by examining Gelman–Rubin–Brooks plots, which showedthe evolution of the shrink factor as the number of MCMC itera-tions increased. Plots were created using the gelman.plot functionin the R package coda (Plummer et al. 2006) with all samples fromthree chains with overdispersed initial values, 5000 samples perchain, and no thinning. Plots were used to identify the number ofiterations at which the shrink factor remained close to 1.0 for allparameters, indicating that chains had converged to stationarydistributions. Results suggested that chains converged to station-ary distribution after about 3000 burn-in samples (Fig. A1).

Total chain length and thinning interval were determined us-ing the run length control diagnostics proposed by Raftery andLewis (1992). We used the raftery.diag function in the R packagecoda with the last 10 000 samples from one chain of 20 000 sam-ples and no thinning to (i) calculate the minimum number ofposterior samples (Nmin) required to estimate the posteriormedian of each parameter to within an accuracy of 1% with95% probability; (ii) determine the dependence factor (I; i.e., extent to

which autocorrelation would inflate the total chain length); and(iii) the total number of samples (N; i.e., chain length beforeburn-in and thinning) required to obtain Nmin samples with noserial autocorrelation. Results suggested that 9604 posterior sam-ples (after burn-in and thinning) would be required to meet ourdesired accuracy and that because of autocorrelation, only onesample out of every eight should be retained (Table A1). To beconservative, we discarded the first 10 000 samples (i.e., burn-in)from a total chain of 110 000 samples and then retained everytenth sample (i.e., thinning). Therefore, all inferences were basedon 10 000 samples that remained after burn-in and thinning. Au-tocorrelation plots (Fig. A2) for each parameter were used to con-firm that consecutive samples were not correlated within the finalposterior sample. Trace plots (Fig. A3) for each parameter wereused to confirm that the posterior distribution was stationary.

ReferencesBrooks, S.P., and Gelman, A. 1998. General methods for monitoring convergence

of iterative simulations. J. Comput. Graph. Stat. 7: 434–455. doi:10.2307/1390675.

Gelman, A., and Rubin, D.B. 1992. Inference from iterative simulation usingmultiple sequences. Stat. Sci. 7: 457–472. doi:10.1214/ss/1177011136.

Plummer, M., Best, N., Cowles, K., and Vines, K. 2006. CODA: Convergence diag-nosis and output analysis for MCMC. RNews, 6: 7–11.

Raftery, A.E., and Lewis, S.M. 1992. One long run with diagnostics: Implementa-tion strategies for Markov chain Monte Carlo. Stat. Sci. 7: 493–497. doi:10.1214/ss/1177011143.

Table A1. Summary of Raftery and Lewis’ (1992) run length control di-agnostics for each parameter with estimated number of posterior sam-ples (Nmin; i.e., chain length after burn-in and thinning) required toachieve desired accuracy, dependence factor (I; i.e., extent to which au-tocorrelation inflates required total chain length), and the total chainlength (N; i.e., chain length before burn-in and thinning) required toobtain Nmin samples with no serial autocorrelation.

Parameter Nmin I N

1 9604 5.94 57 0752 9604 7.52 72 184Deviance 9604 3.79 36 426p2,A 9604 2.41 23 188p3,B 9604 4.12 39 528�1,A,A 9604 2.26 21 720�1,A,C 9604 2.27 21 758�2,A,A 9604 2.66 25 564�2,A,B 9604 4.99 47 898�2,A,C 9604 5.39 51 762�3,B,B 9604 2.25 21 646�3,B,C 9604 2.25 21 642

Holbrook et al. 1723

Published by NRC Research Press

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.

Fig. A1. Gelman–Rubin–Brooks plot showing evolution of Gelman and Rubin’s shrink factor for each parameter as the number of iterationsincreased in a chain of 5000 samples (no thinning).

1724 Can. J. Fish. Aquat. Sci. Vol. 71, 2014

Published by NRC Research Press

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.

Fig. A2. Lagged autocorrelations of each parameter in a chain of 10 000 posterior samples (after burn-in and thinning) from the Bayesianstate-space model.

Holbrook et al. 1725

Published by NRC Research Press

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.

Appendix BThis appendix discusses the sensitivity of parameter estimates

to prior probability distributions in a Bayesian state-space modelused to estimate abundance, spatial distribution, movement, trapcapture, dam passage, and detection probabilities of sea lampreysin the Cheboygan River during 2011.

Prior sensitivity should be evaluated in every Bayesian analysisbecause the choice of prior is subjective and posterior distribu-tions can be influenced by the form of prior distribution. In theBayesian state-space model used to estimate abundance, spatialdistribution, movement, trap capture, dam passage, and detec-tion probabilities of sea lampreys in the Cheboygan River during2011, we used prior probability distributions for three purposes.First, priors were used to apply structural constraints (e.g., fixing

probabilities to zero or one) when transition or detection wasphysically impossible or certain (e.g., �1,A,B = 0; Fig. 1). Our assump-tions of perfect knowledge of those parameters should not re-quire further justification. Second, “flat” priors (e.g., beta(1, 1))were used when we had no prior information about the value of aparameter. In the absence of prior knowledge about a parameter,use of a flat prior is a natural choice, but it is not free of criticismbecause it implies that all values within the parameter space wereequally plausible a priori. This was our choice, however, becauseof the lack of better information. When a flat prior is used, infer-ences are based on the data and the likelihood function becauseposterior probabilities are the product of the prior (with equalprior probability assigned to all values) and the likelihood func-tion. Third, we used priors to specify assumptions needed to esti-

Fig. A3. Trace plots showing sampled value of each parameter for 10 000 posterior samples (after burn-in and thinning) of the Bayesian state-space model.

1726 Can. J. Fish. Aquat. Sci. Vol. 71, 2014

Published by NRC Research Press

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.

mate parameters that were confounded with other parameters(i.e., the data did not contain information needed to separatelyidentify each parameter). Specifically, we assumed that all taggedlampreys passing the dam would have been detected upstream(i.e., p3,A = 1) and that all tagged lampreys caught in the trap weredetected (i.e., p4,B = 1). Those assumptions seemed reasonablebased on ancillary observations, but we considered sensitivity ofresults to those assumptions because they were subjective.

Relaxing assumptions of perfect detectability upstream ofthe dam and in the trap

Detectability upstream of the damInferences from the model were potentially sensitive to as-

sumptions about the probability of detecting a tagged lampreyupstream of the dam (p3,A) because the detection probability up-stream of the dam was confounded with the probability of passingupstream over the dam (�2,A,A). For example, 16 lampreys reachedthe dam but were not detected farther upstream (encounter his-tory AA000). Using only the data, we could not determine howmany of those lampreys passed upstream over the dam but werenot detected by receivers upstream of the dam or ceased migra-tion below the dam. Therefore, an estimate of p3,A was needed toestimate �2,A,A. In the absence of any data to inform estimation ofthose parameters, the model used only the information in theprior probability distributions. In our analysis, inferences werebased on the assumption that all tagged lampreys would havebeen detected by receivers upstream of the dam (i.e., p3,A = 1).Alternatively, we could have used a flat prior probability for bothparameters to represent the assumption that both outcomes were

equally probable a priori or conducted auxiliary field tests to es-timate p3,A and then specified a prior distribution that repre-sented our knowledge of that parameter.

To evaluate sensitivity of results to the assumption that alltagged lampreys would have been detected upstream of the dam,we compared our results, based on our assumption that p3,A = 1,with inferences based on two alternative prior distributions:beta(69, 9) and beta(1, 1). Prior distribution beta(69, 9), with meanof 0.885 and credible interval range of 0.813–0.950 (95% HPDI),was used to represent the assumption that detection probabilityupstream of the dam was equal to the lowest detection probabilityestimated during the study (site 3B; see p3,B, Table 2). Although wedo not expect this “less informative” prior to accurately reflectthe true detection probability at site 3A, it was chosen to be con-servative. Use of this prior also demonstrates how a prior distri-bution might be selected from auxiliary data. For example,beta(69, 9) would have represented our knowledge about p3,A if wehad released 78 tagged lampreys above the dam (to estimate de-tection probability in the absence of any upstream passage) and69 of those were subsequently detected at upstream receivers.Although such a release was not conducted, this prior was used toshow that information would have changed our inferences andmay motivate the inclusion of such auxiliary data in futurestudies.

Inferences would not have changed if we had assumed that theprior probability of detection upstream of the dam (p3,A) was 0.81–0.95 (i.e., beta(69, 9); Fig. B1). Posterior modes of all transitionprobabilities differed by less than 0.005 between the two priors.However, if we had assumed no prior knowledge of the upstream

Fig. B1. Posterior distributions for fish movement probabilities (left column; �2,A,A, �2,A,B, �3,B,B), proportion of the population present at twolocations (middle column; 3,B, 4,B, 3,A), and number of untagged lampreys at three locations (right column; N1,A, N3,A) under three priordistributions for p3,A — the probability that a tagged sea lamprey was detected passing receivers upstream of the Cheboygan River Dam. Forthe coloured version of this figure, refer to the Web site at http://www.nrcresearchpress.com/doi/full/10.1139/cjfas-2013-0581.

Holbrook et al. 1727

Published by NRC Research Press

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.

detection probability (i.e., beta(1, 1)), then we would have inferredthat the detection probability was 0.056 (posterior mode) with95% HPDI range of 0.000–0.853. With no prior information avail-able about the detection probability above the dam, the modelconverged toward a low detection probability because it becameless probable that any of the 16 lampreys with history AA000would have passed the dam undetected. This inference does notseem plausible given our observations that receivers upstream ofthe dam consistently detected sentinel tags throughout the studyand performed better than acoustic receivers downstream of thedam, where estimated detection probabilities were 0.88 and 0.98(posterior modes). Nonetheless, only the probability of dam pas-sage (�2,A,A), the proportion of the population that escaped up-stream (3,A), and the number of individuals that escapedupstream (N3,A) would have been affected by use of a flat prior forp3,A (Fig. B1). The point estimate (posterior mode) of �2,A,A wouldhave only increased from 0.001 to 0.005; however, the upper limitof the credible interval would have increased from 0.020 to 0.077(95% HPDI). As a result, the 95% HPDI for 3,A would haveincreased from 0%–1.9% of the population to 0%–7.5% of thepopulation escaping above the dam, and the upper limit of95% HPDI for N3,A would have increased from 514 lampreys to1943 lampreys that escaped above the dam.

Detectability in the trapInferences from the model were potentially sensitive to as-

sumptions about the probability of detecting a tagged lamprey inthe trap (p4,B) because the detection probability in the trap wasconfounded with the probability of trap capture (�3,B,B). For exam-

ple, 47 lampreys that reached the spill basin were not detected inthe trap (encounter history AAB00). Using only the data, we couldnot determine how many of those lampreys were caught in thetrap but were not detected by trap personnel or were never caughtin the trap. Therefore, the probability of detecting a lamprey inthe trap (p4,B) was needed to estimate the probability of enteringthe trap (�3,B,B). In the absence of any data to inform estimationof those parameters, the model used only the information in theprior probability distributions. In our analysis, inferences werebased on the assumption that all tagged lampreys were detectedby trap personnel (p4,B = 1). Alternatively, we could have used a flatprior probability for both parameters to represent the assumptionthat both outcomes were equally probable or conducted auxiliaryfield tests to estimate p4,B and then used a prior distribution thatrepresented our knowledge of that parameter.

To evaluate sensitivity of results to the assumption that alltagged lampreys were detected in the trap, we compared our re-sults, based on our assumption of perfect prior knowledge (p4,B = 1),with inferences based on two alternative prior distributions:beta(90, 10) and beta(1, 1). Prior distribution beta(90, 10), withmean of 0.900 and credible interval range of 0.840–0.954(95% HPDI), was used to represent a hypothetical condition whereabout one of every ten acoustic-tagged fish caught in the trap wasnot identified by trap personnel. We consider this “less informa-tive” prior to be conservative (i.e., the true detection probability oftagged lampreys in the trap was probably higher) because tagdetection equipment was tested daily and occasional “double-scanning” of daily catches has never revealed any tagged lampreys

Fig. B2. Posterior distributions for fish movement probabilities (left column; �2,A,A, �2,A,B, �3,B,B), proportion of the population present at twolocations (middle column; 3,B, 4,B, 3,A), and number of untagged lampreys at three locations (right column; N1,A, N3,A) under three priordistributions for p4,B — the probability that a tagged sea lamprey was identified in the Cheboygan River trap. For the coloured version of thisfigure, refer to the Web site at http://www.nrcresearchpress.com/doi/full/10.1139/cjfas-2013-0581.

1728 Can. J. Fish. Aquat. Sci. Vol. 71, 2014

Published by NRC Research Press

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.

missed by coded-wire tag detectors. Use of this prior also demon-strates how a prior distribution might have been selected fromauxiliary data. For example, beta(90, 10) would have representedour knowledge about p4,B if we had placed 100 tagged lampreys inthe trap and 90 of those were subsequently identified as tagged bytrap personnel. Although we did not place any tagged lampreys inthe trap to estimate the detection probability, this prior was usedto show how that information would have changed our inferencesand may motivate the inclusion of such auxiliary data in futurestudies.

As expected, the probability of entering the trap from the spillbasin (�3,B,B; i.e., local trap efficiency) was sensitive to our assump-tion of perfect detectability in the trap (Fig. B2). If we had assumedthat the true detection probability was in the range 0.84–0.95 (i.e.,p4,B � beta(90, 10)), then our point estimate of �3,B,B would haveincreased from 0.602 to 0.673 and the credible interval for thatparameter would have changed from 0.516–0.685 to 0.563–0.775.Posterior modes of all other transition probabilities differed byless than 0.004 between the two priors. Increased trap efficiencywould not have changed our estimate of the proportion of the

population vulnerable to the trap (3,B), but our estimate of pro-portion of the population caught in the trap (4,B; i.e., exploita-tion rate) would have increased from 0.542 to 0.585 (posteriormode) and the credible interval for that parameter would havechanged from 0.457–0.613 to 0.503–0.696. A larger estimate of�3,B,B would have also resulted in smaller estimates of spawnerabundance (Fig. B2). Credible intervals (95% HPDI) for the numberof spawners that entered the river (N1,A) would have changed from21 828 – 29 299 to 19 426 – 26 878, and the upper limit of thecredible interval for the number of lampreys that escaped abovethe dam (N3,A) would have decreased from 514 to 449. If we hadassumed no prior knowledge of p4,B (i.e., p4,B � beta(1, 1)), then wewould have inferred that trap efficiency (�3,B,B) was 0.679 (poste-rior mode) with a credible interval (95% HPDI) range of 0.594–1.00.This increase in both the point estimate and uncertainty of �3,B,Bwould have resulted in a further increase of the point estimateand uncertainty of the exploitation rate (4,B) and decreased esti-mates of abundance, but would not have changed our estimate ofthe proportion of the population vulnerable to the trap (3,B;Fig. 2B).

Holbrook et al. 1729

Published by NRC Research Press

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y M

ICH

IGA

N S

TA

TE

UN

IVE

RSI

TY

on

10/3

0/14

For

pers

onal

use

onl

y.